Invoke the File Browser inside an Add-On but wait with the rest of the script until it's closed
The solution was very simple at the end. Everything what needs to be done with the data which was loaded by the ImportHelper should be part of the execute method of this class.
The invoke method (in my above script) was not needed at the end:
class ImportFBXData(Operator, ImportHelper):
    bl_idname = &quot;import_fbx.read_data&quot;  
    bl_label = &quot;Import FBX&quot;
    
    filter_glob: StringProperty(
        default='*.fbx',
        options={'HIDDEN'})
    
    def execute(self, context):
        # Everything what needs to be done before opening the file browser 
        new_fbx_collection = bpy.data.collections.new(&quot;New COllection&quot;)
        bpy.context.scene.collection.children.link(new_fbx_collection)

        # Open the File Browser and Import the FBX File
        filename, extension = os.path.splitext(self.filepath)
        bpy.ops.import_scene.fbx(filepath = self.filepath)
        
        # Everything what needs to be done after the file browser 
        ...
        obs = [o for o in bpy.data.collections[context.collection.name].all_objects if not o.parent and len(o.children)]
        for o in obs:
            ...
    return {'FINISHED'}

This way my Add-On works perfectly.